The path of a general cycloid is parameterized by \(x=r(t-\sin{t})\) and \(y=r(1-\cos{t})\)

(a) Find the length of one arch of the cycloid.

(b) Let \(L\) be the length you found in part (a). One end \(A\) of a string of length \(\dfrac{L}{2}\) is tied at \((0, 0)\), and the other end \(B\) is at \((\pi r, 2r)\), wrapping around one-half of one arch of the cycloid. 

Prove that as the string is unwrapped, the point \(B\) traces a path that is parameterized by \(x=r(t+\sin{t})\) and \(y=r(3+\cos{t})\). (This path is in fact congruent to an arch of the original cycloid.)


Hi, it would really help if anyone could help with part (b) as I have no idea where to start. For part (a) I got 8r for length L.

 Jul 21, 2022

This is a very easy problem.  Just use the parametric form of the cycloid.

 Jul 22, 2022

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